altium autoroute jumper
No, Altium Designer does not support single layer autorouting with jumpers. You would have to set up the routing rules for a two sided board, with routing cost adjusted to favor one side. You would then have to manually place jumpers.
Likewise, Allegro does not support single sided boards with Specctra autorouting. You would have to do the same as suggested above.
I'm sure you will get a flood of recommendations for EDA software that can do that sort of thing. There are several hobby level PCB design packages that do auto jumpers (DipTrace and Number One Systems for example).
PADS has jumpers that can be manually placed and the autorouter will recognize them, but it doesn't have the ability to generate jumpers automatically from the autorouter.